Why won't Lasso or Box Select select all of the faces in the region?
There are several ways to select every vertice of the mesh not changing view angle in the scene.

Limit Selection to Visible

Uncheck Limit selection to visible. 



If it's checked, it won't be possible to select vertices which aren't pointed to the viewer (like in your case). 
If it isn't checked, when selecting either with Box select or with RMB (doesn't matter how) vertices on the other side of the mesh will be added to selection. 

Wireframe viewport shading

Use Wireframe shading mode, accessible via shortcut Z while in 3D Viewport.



Selecting connected geometry


  Select parts of a mesh connected to already selected elements. 


Use Ctrl+L shortcut for that or from menu Select > Linked. 


  You can also select linked data directly under the cursor.


Press L to select linked data and Shift+L to deselect data which is located under cursor.
Read more about selecting linked in the wiki page.